Lifting of supersingular points on X0 (pr) and lower bound of ramification index

Abstract

Let K be a finite extension of (= the maximal unramified extension of Qp) of degree eK, its integer ring, p a rational prime and r a positive integer. If there exists a one parameter formal group defined over whose reduction is of height 2 with a cyclic subgroup V of order pr defined over , then .We apply this result to a criterion for non-existence of Q-rational point of . (This criterion is Momose’s theorem in [14] except for the cases p = 5 and p = 13, but our new proof does not require defining equations of modular curves except for the case p = 2.)